Collocations for university - noun

elite, leading, major, prestigious, top
ancient (esp. BrE)
modern, new, red-brick (BrE)
Catholic, private, public (all esp. AmE)
land-grant (AmE)
local, provincial (BrE), state (AmE)
research

attend, go to, study at
He goes to Princeton University.

apply to
I applied to three universities.
I'm applying to university this year. (BrE)

enter
King entered Montana State University in 2002.
Young people may be deterred from entering university. (BrE)

finish (BrE), graduate from, leave
She graduated from the University of Michigan in 1999.
I graduated from university last year. (BrE)

lecturer (esp. BrE), professor, researcher, staff, teacher
graduate, student
course, degree, education
research
fees (BrE), tuition (AmE)
the government's decision to introduce university top-up fees
those who can afford university tuition

funding (esp. BrE)
term (BrE), year
vacation
admission, admissions, entrance, entry (esp. BrE), place (esp. BrE)
a university entrance exam
There is stiff competition for university places.

chair
His aim was to obtain a university chair.

department
accommodation (BrE), buildings, campus, library
chancellor, vice chancellor
dean, president, provost (all in the US)
administration, authorities (BrE)
administrator, official (both esp. AmE)
faculty (= those who teach), staff (= those who teach and those who provide services)
staff (= those who provide services)
hospital, laboratory, medical centre/center, press
This book was published by Oxford University Press.

level
All the staff are educated to university level.

system
life (esp. BrE)
the pressures of university life

days, years (both esp. BrE)
I often think of my university days.

town

at (the) university
She teaches botany at Syracuse University.
She's at university, studying engineering. (BrE)